A208206 a(n)=(a(n-1)^2*a(n-2)+1)/a(n-3) with a(0)=a(1)=a(2)=1.

1, 1, 1, 2, 5, 51, 6503, 431347892, 23724602128927104843, 37334625705205335653803036700733450756576803

Offset: 0

Author

Matthew C. Russell, Apr 23 2012

Keywords

• nonn

Comments

This is the case a=1, b=2, y(0)=y(1)=y(2)=1 of the recurrence shown in the Example 3.2 of "The Laurent phenomenon" (see Link lines, p. 10).

The next term (a(10)) has 98 digits. - Harvey P. Dale, Oct 04 2014

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..12
• Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, arXiv:math/0104241v1 [math.CO] (2001), Advances in Applied Mathematics 28 (2002), 119-144.

Crossrefs

Cf. A005246, A208202, A208207, A208209.

Programs

Maple

a:=proc(n) if n<3 then return 1: fi: return (a(n-1)^2*a(n-2)+1)/a(n-3): end: seq(a(i),i=0..10);

Mathematica

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(a[n-1]^2 a[n-2]+1)/a[n-3]},a,{n,10}] (* Harvey P. Dale, Oct 04 2014 *)

Formula

From Vaclav Kotesovec, May 20 2015: (Start)

a(n) ~ c1^(d1^n) * c2^(d2^n) * c3^(d3^n), where

d1 = -0.80193773580483825247220463901489010233183832426371430010712484639886484...

d2 = 0.554958132087371191422194871006410481067288862470910089376025968205157535...

d3 = 2.246979603717467061050009768008479621264549461792804210731098878193707304...

are the roots of the equation d^3 + 1 = 2*d^2 + d and

c1 = 0.874335057499939749225491691816700793966151250175012051621456437468590379...

c2 = 0.402356411273897640287204171338236092104516307383060911032953286637247174...

c3 = 1.071117422488325114038954501945557033632156032599675833309484054582086570...

(End)